2 Cracking of concrete structures
3.7 Sample calculation of crack width in a special case
In this section, the calculation of the maximum crack width is performed to a special case of a reinforced concrete structure according to each method. An example structure is 2000mm thick and heavily reinforced concrete wall that demonstrates the wall of a containment structure enclosing a nuclear reactor. As a simplification we will consider it as a beam of a width of 1000mm. Reinforcement is embedded into four layers, in which each consists of two orthogonally set 40mm diameter bars with bar spacing 300mm. Spacing of two parallel bars in different layers is 100mm. A principle sketch of the cross section is shown in the Figure 3.6, in which tension bars are shown at the plane of the figure. The thickness of the concrete cover c used in the calculations is now 100mm. Concrete class is C35/45, and reinforcing steel B500C1. The structure is sub- jected to pure bending.
Figure 3.6. Cross section of the special case structure.
Next, we are calculating the cracking moment for this cross-section in order to evaluate steel stress caused by pure bending in case cracking occurs. The cracking moment Mcr
can be evaluated from the geometry with assuming the effective tensile strength fct,eff as
fctm (=3,2MPa for concrete C35/45):
= 3,2 ( ) = 2133,3
When the cracking moment is evaluated, we can assume in the calculations the structure to be subjected to bending moment of 8000kNm in order to ensure the cracking can oc- cur.
Calculation according to EC2 3.7.1
The progress of the crack width calculation is similar to that of the simple case. It starts with calculating the position of the neutral axis. At first, the following values need to be calculated:
= = 200000
= = 1730167551000 = 0,00969
= 5,869 0,00969 = 0,0568
= 0,0568 + 0,0568(2 + 0,0568) 0,399
Now, the position of the neutral axis is:
= =
+ 1 d = 0,399
1,399 1730 = 493
After that, the height of the effective tension area is:
= min 2,5(2000 1730);2000 4933 ;20002 = 502mm
The effective tension area Ac,eff is then: = 502267mm
Now that we know the position of the neutral axis, we can evaluate the steel stress caused by the chosen bending moment M (=8000kNm) which is greater than cracking moment. The average steel stress in the bars equals:
= ( 3)= 8000
16755 (1730 493 3)= 305
The maximum steel stress in the bar layer closest to concrete surface in the tension area is obtained from the figure 3.7 due to geometry when the average stress in the bars equals 305MPa:
= 305 (1880(1730 493493 )) = 342
Prior to calculate the maximum crack spacing, p,eff is calculated from the expression 3.13:
, =502267mm = 0,033416755mm
Now, the maximum crack spacing can be evaluated from the expression (3.2). The fac- tors used in the equation are:
k1 = 0,8 (bars are assumed to have good bond characteristics)
k2 = 0,5 (in case of bending)
k3 = 3,4 (recommended value)
k4 = 0,425 (recommended value)
, = 3,4 100 + 0,8 0,5 0,4250,0334 = 52340
When the value 0,4 is given to the factor kt due to long term loading, the difference be-
tween the main strains of steel and concrete can be calculated from the expression 3.4:
=342MPa 0,4 3,2MPa200000MPa0,0334 (1 + 5,869 0,0334)= 0,00148
Finally, the crack width can be calculated according to the expression 3.1: = 523 0,00148 = 0,775
Calculation according to RakMK 3.7.2
At first, the ratio between the area of reinforcing steel and tension area of the concrete is calculated. The height of the area reaches the level, which is at the distance of 7.5 times the bar diameter from the innermost bar.
A = 1000mm 100mm + + 3 100mm + 7,5 40mm = 720000mm = 720000mm = 0,02316755mm
The tensile stress in the bars at time of the first crack development needs to be calculat- ed next. The position of the neutral axis was already calculated in subsection 3.7.1. The tensile stress sr can be calculated from the expression 3.17:
= 1,7 1000mm (2000mm)6 2,2MPa
(1730mm 13 493mm) 16755mm = 95,0MPa
Now, the average steel strain and the final crack width can be calculated from the ex- pression 3.5 and 3.6:
= 200000342 25 0,0851 95,0342 = 0,00165
= 0,00165 3,5 100 + 0,085 400,023 0,818
Calculation according to DIN 1045 3.7.3
The calculation according to DIN begins, again, with calculating the effective tension area from the equation 3.14:
= 1000 675 = 675000
After that, the reinforcement level can be evaluated from the expression 3.15: 16755
675000 = 0,0248
Now, the maximum crack spacing sr,max and the average strain difference sm - cm can
be obtained from the expression 3.8 and 3.9:
=3,6 0,0248 <40 3423,6 3,240 = 447,6
=342 0,4 3,20,0248 (1 + 5,869 0,0248)200000 < 0,6200000342
= 0,00142
As a final result, the crack width for this special case according to the DIN 1045 can be evaluated from the equation 3.7:
= 447,6 0,00142 = 0,633
Calculation according to ACI 318 3.7.4
Lastly, the maximum crack width in special case is calculated by ACI 318. Tension area is calculated from the equation 3.16 at first:
= (78,74 68,5)39,37 39,37
11,81 4
60,45
When taking into account the previously calculated position of the neutral axis 19.3in, parameter gets the value of:
=78,74in 19,42in68,1in 19,42in = 1,218
When the tensile stress in the bars is assumed to be 342MPa that equals 49602 pounds per square inch, the crack width in inches can be calculated from the equation 10:
= 0,076 1,218 49603 (78,74 68,1) 62,78 = 0,0401in (1,027mm)
3.8 Summary of the special case results
The results of the crack widths calculated for the special case are summarized in this section in order to obtain information about crack widths in uncommon cases.
Table 3.5. Crack width results in the special case.
EC2 RakMK DIN ACI Crack width (mm) 0,775 0,818 0,633 1,027 Difference compared to EC2 0 % 5 % -18 % 33 %
As it can be noticed the maximum crack widths are way out of the limiting values set for the maximum crack width 0,2mm, 0,3mm or 0,4mm, depending on the exposure class. Calculation by DIN seems to result the lowest crack width values. Value accord- ing to EC2, almost, equals to that of the RakMK, which is notable. As stated in previous section 3.5, in case of normal structure crack width values calculated by EC2 are about 30% lesser than values according to RakMK. The effect of steel stress and cover thick- ness is explored by performing two different tables, in which the mentioned parameters are varied.
Table 3.6. Crack widths in the special case with different steel stresses.
The maximum steel stress
Crack width
EC2 RakMK DIN 1045 ACI 318 342 MPa 0,775 0,818 (5 %) 0,633 (-18 %) 1,027 (33 %) 300 MPa 0,665 0,709 (7 %) 0,539 (-19 %) 0,901 (32 %) 260 MPa 0,506 0,542 (7 %) 0,387 (-24 %) 0,691 (37 %) 220 MPa 0,409 0,447 (9 %) 0,307 (-25 %) 0,585 (43 %)
Table 3.7. Crack widths in the special case with different cover thickness.
Cover thickness Crack width
EC2 RakMK DIN 1045 ACI 318 50 mm 0,470 0,448 (-5 %) 0,453 (-4 %) 0,746 (59 %) 60 mm 0,513 0,501 (-2 %) 0,470 (-8 %) 0,773 (51 %) 70 mm 0,556 0,554 (0 %) 0,486 (-13 %) 0,801 (44 %) 80 mm 0,598 0,606 (1 %) 0,502 (-16 %) 0,829 (39 %) 90 mm 0,640 0,658 (3 %) 0,517 (-19 %) 0,857 (34 %) 100 mm 0,775 0,818 (5 %) 0,633 (-18 %) 1,027 (33 %)